1
High-Throughput Data Detection for Massive
MU-MIMO-OFDM using Coordinate Descent
Michael Wu, Chris Dick, Joseph R. Cavallaro, and Christoph Studer
Abstract—Data detection in massive multi-user (MU) multipleinput multiple-output (MIMO) wireless systems is among the
most critical tasks due to the excessively high implementation
complexity. In this paper, we propose a novel, equalization-based
soft-output data-detection algorithm and corresponding reference
FPGA designs for wideband massive MU-MIMO systems that
use orthogonal frequency-division multiplexing (OFDM). Our
data-detection algorithm performs approximate minimum meansquare error (MMSE) or box-constrained equalization using
coordinate descent. We deploy a variety of algorithm-level optimizations that enable near-optimal error-rate performance at
low implementation complexity, even for systems with hundreds
of base-station (BS) antennas and thousands of subcarriers. We
design a parallel VLSI architecture that uses pipeline interleaving
and can be parametrized at design time to support various
antenna configurations. We develop reference FPGA designs for
massive MU-MIMO-OFDM systems and provide an extensive
comparison to existing designs in terms of implementation
complexity, throughput, and error-rate performance. For a 128
BS antenna, 8 user massive MU-MIMO-OFDM system, our
FPGA design outperforms the next-best implementation by more
than 2.6× in terms of throughput per FPGA look-up tables.
with hundreds of antenna elements and thousands of subcarriers,
novel algorithms and dedicated hardware implementations on
field-programmable gate arrays (FPGAs) or application specific
integrated circuits (ASICs) are necessary.
During recent years, various data-detection algorithms [8],
[9] and dedicated hardware implementations have been proposed for massive MU-MIMO systems [7], [10]–[13]. All of
the existing hardware implementations, however, are either
unable to achieve the high throughputs offered by future
wideband massive MU-MIMO systems [7], [12], [13], or
exhibit excessive hardware complexity [11]. Furthermore, the
hardware implementations in [7], [11] only support singlecarrier frequency-division multiple-access (SC-FDMA). As
demonstrated in [14], however, orthogonal frequency-division
multiplexing (OFDM) enables (often significantly) less complex
baseband processing1 , which may be a critical design factor
for wideband massive MU-MIMO systems with hundreds of
BS antennas and thousands of subcarriers.
Index Terms—Coordinate descent, equalization, FPGA design,
massive multi-user (MU) MIMO, orthogonal frequency-division
multiplexing (OFDM), soft-output data detection.
A. Contributions
I. I NTRODUCTION
ASSIVE multi-user (MU) multiple-input multipleoutput (MIMO) technology promises significant improvements in terms of spectral efficiency, coverage, and
range compared to traditional, small-scale MIMO [2]–[5].
In fact, massive MU-MIMO is commonly believed to be
one of the key technologies for future fifth-generation (5G)
wireless systems [6]. The idea underlying this technology is to
equip the base-station (BS) with hundreds of antenna elements
while communicating with tens of user terminals concurrently
and within the same time-frequency resource. However, the
large dimensionality of the data detection problem faced in
the uplink (where users communicate to the BS), results in
excessively high implementation complexity at the BS (see,
e.g., [7] and the references therein). Hence, to reduce the
implementation costs while enabling throughputs in the Gb/s
regime for practical wideband massive MU-MIMO systems
M
MW and JRC are with the Department of ECE, Rice University, Houston,
TX; e-mail: {mbw2,cavallar}@rice.edu
MW and CD are with Xilinx Inc., San Jose, CA; e-mail: {miwu,
chris.dick}@xilinx.com
CS is with the School of ECE, Cornell University, Ithaca, NY; e-mail:
[email protected]
A short version of this paper for a single-carrier frequency-division multiple
access (SC-FDMA) massive MU-MIMO systems has been presented at the
IEEE International Symposium on Circuits and Systems (ISCAS) [1].
We propose a new, low-complexity soft-output data-detection
algorithm and a corresponding high-throughput FPGA design
for massive MU-MIMO wireless systems that use OFDM. Our
algorithm, referred to as optimized coordinate descent (OCD),
performs approximate minimum mean-square error (MMSE)
or box-constrained equalization, which enables near maximumlikelihood (ML) soft-output data detection performance in
massive MU-MIMO systems with a large BS-to-user-antenna
ratio. We develop a corresponding high-throughput VLSI
architecture with a deep and interleaved pipeline, which can
be parametrized at design time to support various BS and user
antenna configurations. The algorithmic regularity of OCD and
the fact that preprocessing can be implemented at minimum
hardware overhead enables high-throughput VLSI designs that
require lower complexity than state-of-the-art designs, even
for systems with hundreds of BS antennas and thousands of
subcarriers. To demonstrate the advantages of OCD compared
to existing massive MU-MIMO data-detector designs in terms
1 SC-FDMA typically generates baseband signals with a lower dynamic
range, but the receiver must perform an additional frequency-to-time conversion
(compared to OFDM). This additional conversion step requires one to separate
equalization (that is usually carried out in the frequency domain per subcarrier)
and data detection (that must be carried out in the time domain). This separation
prevents the use of powerful, non-linear equalization methods [15], such as the
box-constrained detector proposed in this paper. OFDM, in contrast, causes
a slightly higher dynamic range, but requires only one time-to-frequency
conversion and enables non-linear data-detection methods that operate directly
in the frequency domain on a per-subcarrier basis [14].
2
of throughput, hardware complexity, and error-rate performance,
we provide implementation results on a Xilinx Virtex-7 FPGA.
B. Notation
Boldface lowercase and boldface uppercase letters stand for
column vectors and matrices, respectively. For a matrix A,
we denote its hermitian transpose by AH . We use ak,` for
the entry in the kth row and `th column of the matrix A; the
kth entry of a column vector a is denoted bypaP
k = [a]k . The
2
`2 -norm of a vector a is defined as kak2 =
k |ak | . The
real part of a complex number a is <{a}. Sets are denoted by
uppercase calligraphic letters; the cardinality of a set A is |A|.
The expectation operator is designated by E[·].
C. Paper Outline
The rest of the paper is organized as follows. Section II
introduces the massive MU-MIMO-OFDM system model and
describes data detection using MMSE and box-constrained
equalization. Section III details our OCD algorithm and shows
error-rate simulation results. Section IV and Section V describe
our VLSI architecture and shows FPGA implementation results,
respectively. We conclude in Section VI.
II. S YSTEM M ODEL AND DATA D ETECTION
This section introduces the considered OFDM-based uplink
model and summarizes efficient methods for linear MMSE and
box-constrained soft-output data detection.
B. Equalization-based Data Detection
For the model in (1), optimal data detection in terms of
minimizing the symbol error-rate is accomplished by solving
the maximum-likelihood (ML) problem [18]
2
s̃ML
w = arg min kyw − Hw zk2 .
(2)
z∈O U
Unfortunately, solving (2) exactly for massive MU-MIMO
systems quickly results in prohibitive complexity, even with
the best-known sphere-decoding algorithms [19]. Equalizationbased data detection algorithms [18] enable one to find
approximate solutions to the ML problem at low computational
complexity. Virtually all linear as well as non-linear equalization methods relax the finite-alphabet constraint z ∈ OU in (2),
which enables the efficient computation of an estimate s̃ that is
(hopefully) close to the ML solution. The estimate s̃ can then
either be sliced element-wise onto the nearest constellation
point in O as follows:
ŝi = arg min |[s̃]i − z|,
i = 1, . . . , U,
(3)
z∈O
which is known as hard-output data detection, or used to
compute reliability information for each transmitted bit in the
form of log-likelihood ratio (LLR) values (see Section II-E),
which is known as soft-output data detection [20], [21].
C. Linear MMSE Equalization
The most common equalization-based data detection algorithm is linear MMSE data detection [18], [20]. This
method was shown to enable FPGA and ASIC designs that
are able to achieve high throughput in massive MU-MIMO
systems [7]. Furthermore, for systems with large BS-to-user
antenna ratios δ = B/U (e.g., two or larger), linear detectors
are able to achieve near-ML error-rate performance [3]–[5].
The key idea of MMSE data detection is to relax the
constraint z ∈ OU in the ML problem (2) to the U -dimensional
complex space z ∈ CU , and to include a quadratic penalty
function. In particular, MMSE equalization solves the following
regularized least-squares problem [10], [22]:
A. OFDM-based Uplink System Model
We consider a massive MU-MIMO-OFDM uplink system,
where U single-antenna user terminals send data simultaneously
to a BS with B U antennas over W subcarriers. Each user
i = 1, . . . , U encodes its own bit stream (using a forward
error-correction scheme) and maps the generated coded bits
onto constellation points in a finite set O (e.g., 64-QAM
using a Gray
mapping rule), with unit average transmit power,
i.e., E |s|2 = 1 with s ∈ O, and Q = log2 |O| bits per
constellation point. The resulting W frequency-domain symbols
(i)
(i)
{s1 , . . . , sW } are then transformed into the time domain (TD)
s̃MMSE
= arg min kyw − Hw zk22 + N0 kzk22 .
(4)
w
using an inverse discrete Fourier transform (DFT) [16]. After
z∈CU
prepending the cyclic prefix, all users transmit their TD signals
Since the objective function in (4) is quadratic in z, the MMSE
over the frequency-selective wireless channel at the same time.
equalization problem has a closed-form solution.
After removing the cyclic prefixes, the TD signals received at
An explicit solution to (4) can be computed as follows. First,
each BS antenna are transformed back to the FD using a DFT.
compute the regularized Gram matrix Aw = Gw + N0 IU with
For the sake of simplicity, we assume a sufficiently long cyclic
MF
H
Gw = HH
w Hw and the matched filter vector s̃w = Hw yw .
prefix, perfect synchronization, and that perfect channel-state
Then, the MMSE estimate in (4) is computed as
information (CSI) has been acquired via pilot-based training.2
MF
s̃MMSE
= A−1
(5)
Under these assumptions, the FD input-output relation on the
w
w s̃w .
wth subcarrier is commonly modeled as [17]
While this closed-form approach was shown to be efficient
yw = Hw sw + nw , w = 1, . . . , W,
(1) for traditional, small-scale MIMO systems (e.g., with four
antennas at both ends of the wireless link) [21], computing the
where yw ∈ CB is the associated received FD vector, Hw ∈
regularized Gram matrix Aw and its inverse A−1
w quickly
CB×U is the channel matrix, sw ∈ OU contains the symbols
results in prohibitive complexity in massive MU-MIMO
(i)
transmitted by all U users, i.e., [sw ]i = sw refers to the symbol
systems with hundreds of BS antennas [11]. In Section III,
transmitted by user i over subcarrier w, and nw ∈ CU models
we present a computationally-efficient equalization algorithm
thermal noise as i.i.d. complex circularly-symmetric Gaussian
that directly solves (4) in a hardware efficient way, which
vector with variance N0 per complex entry.
avoids expensive calculations such as the computation of the
2 These assumptions are common in the MIMO-OFDM literature [16].
regularized Gram matrix Aw and its inverse A−1
w .
3
D. Non-Linear Box-Constrained (BOX) Equalization
While linear equalization methods are the most common
approach in the MIMO literature, a few non-linear equalizers
have recently emerged and shown to outperform linear methods
in terms of error-rate performance [23]. A promising non-linear
equalization method, referred to as box-constrained equalization
(short BOX equalization) [24]–[26], relaxes the constraint
z ∈ OU to the convex polytope CO around the constellation
set O, which is formally defined as follows:


|O|
|O|
X

X
CO =
αi si | (αi ≥ 0, αi ∈ R, ∀i) ∧
αi = 1 . (6)


i=1
i=1
For example, the convex polytope CQPSK for QPSK with3
O = {+1 + j, +1 − j, −1 + j, −1 − j}
(7)
is given by CQPSK = {xR + jxI : xR , xI ∈ [−1, +1]} with
j 2 = −1; this is simply a box with radius 1 around the
square constellation (thus the name BOX equalization). For
higher-order QAM alphabets, such as 16-QAM or 64-QAM,
we have CO = {xR + jxI : xR , xI ∈ [−α, +α]}, where
α = maxa∈O <{a} is the radius of the tightest box around
the square constellation.
BOX equalization solves the following relaxed version of
the ML problem in (2):
s̃BOX
= arg min kyw − Hw zk22 .
w
(8)
U
z∈CO
where the sets Ob0 and Ob1 contain the constellation symbols
for which the bth bit is 0 and 1, respectively. For explicit
MMSE detection, i.e., the approach discussed in Section II-C
that computes A−1
w , the post-equalization signal-to-noise-andinterference-ratio (SINR) ρw,i and the channel gain µw,i can
be calculated exactly and in the following efficient way [21].
The SINR is calculated as ρw,i = µw,i /(1 − µw,i ) and the
channel gain is µw,i = [Aw ]H
i [Gw ]i , where [Aw ]i is the ith
row of A−1
w and [Gw ]i is the ith column of Gw .
However, for BOX equalization in Section II-D, as well
as for data detection algorithms that implicitly solve the
MMSE detection problem (4), no efficient methods that
exactly compute the SINR ρw,i are known—this prevents
a straightforward computation of the LLR values in (9). In
Section III-C, we propose an approximate way to generate ρw,i
and µw,i , which enables us to compute approximate LLR values
for such linear and non-linear equalizers.
III. FAST E QUALIZATION VIA C OORDINATE D ESCENT
While the solution to the implicit MMSE problem (4)
can be computed (exactly or approximately) at moderate
complexity using iterative conjugate gradient (CG) or GaussSeidel (GS) methods, see, e.g., [9], [13], [22], corresponding
VLSI designs [10], [13] are unable to achieve high throughput,
mainly due to a fairly complex algorithm structure, stringent
data dependencies, or the need for high arithmetic precision. We
next propose an alternative method to solve both the MMSE
equalization (4) and BOX equalizaton (8) problems at low
complexity and in a hardware friendly way.
Since this equalization problem (8) is convex, it can be solved
exactly using well-established numerical methods from convex
optimization [27]. Furthermore, as shown recently in [23], [26],
the BOX equalizer exhibits near-ML error-rate performance in A. Coordinate Descent (CD)
the large-antenna limit, where we fix the BS-to-user antenna
Coordinate descent (CD) [29] is a well-established iterative
ratio δ = B/U so that δ > 1/2 and by letting B → ∞. In
addition, the BOX equalizer does only need knowledge of the framework to exactly or approximately solve a large number
of convex optimization problems using a series of simple,
transmit constellation O but not of the noise variance N0 .
Unfortunately, solving (8) exactly with conventional interior- coordinate-wise updates. We first define the following function:
point methods results in prohibitive complexity and requires
f (z1 , . . . , zU ) = f (z) = kyw − Hw zk22 + g(z),
(10)
high numerical precision, which prevents efficient hardware
designs that use finite precision (fixed-point) arithmetic. In
order to solve (8) at low complexity and in a hardware efficient where g(z) is a convex regularizer. It is now important to
realize that both equalization problems (4) and (8) are special
way, we propose a new algorithm in Section III.
cases when minimizing (10). In fact, by setting g MMSE (z) =
N0 kzk22 , minimizing (10) is equivalent to solving the MMSE
E. Soft-Output Data Detection
equalization problem (4). By setting g BOX (z) = χ(z ∈ CO ),
From MMSE and BOX equalization, hard-output estimates where χ(z ∈ CO ) denotes the characteristic function that is
can easily be obtained by element-wise slicing of the entries of zero if z ∈ CO and infinity otherwise, minimizing (10) is
s̃MMSE
and s̃BOX
onto the nearest constellation point as in (3), equivalent to solving the BOX equalization problem (8). CDw
w
respectively. In systems that use forward error-correction, how- based equalization simply minimizes the function f (z1 , . . . , zU )
(or coordinate) zu , u =
ever, one is generally interested in soft-output detection [28]. in (10) sequentially for each variable
4
MMSE
1,
.
.
.
,
U
,
in
a
round-robin
fashion.
For
more details on CD,
From MMSE equalization where s̃w = s̃w
, LLR values are
see
[29],
[30]
and
the
references
therein.
We next detail the
typically computed via the max-log approximation [21]
CD
algorithms
for
MMSE
and
BOX
equalization.
2
2 !
[s̃w ]i
[s̃w ]i
Lw,i,b = ρw,i min0 − a − min1 − a , (9)
4 The performance of CD can often be improved by using a carefully-selected
a∈Ob µw,i
a∈Ob µw,i
3 We
note that this constellation is not normalized to unit expected power.
variable-update order [29]; our own experiments have shown that for MMSE
and BOX equalization, a simple round-robin update scheme performs well
and is easier to implement.
4
1) CD-based MMSE Equalization: Assume we want to
find the uth optimum value zu for the MMSE equalization
problem (4), i.e., we seek to compute the solution to
ẑu = arg min kyw − Hw zk22 + N0 kzk22 ,
(11)
zu ∈C
where we hold all other values zj , ∀j 6= u, fixed. Since this
is a quadratic problem, we can solve it in closed form by
setting the gradient of the function (10) with respect to the uth
component to zero:
0 = ∇u f (z) = hH
u (Hz − y) + N0 zu .
(12)
P
By decomposing Hz = hu zu + j6=u hj zj , we can solve (12)
for zu to obtain the following closed-form expression:


X
1
ẑu =
hH y −
hj zj .
(13)
khu k22 + N0 u
j6=u
This expression is exactly the CD update rule for the uth entry
of z. For every iteration, we can compute (13) sequentially
for each user u = 1, . . . , U , where we immediately re-use the
new result ẑu for the uth user in subsequent steps. We repeat
this procedure for a total number of K iterations in order to
obtain an estimate for s̃MMSE = z(K) , where z(K) is the end
result of the above-described iterative process.
2) CD-based BOX Equalization: Analogously to CD-based
MMSE equalization, we can derive the update rule for the
BOX equalization problem (8). Even though the characteristic
function g BOX (z) = χ(z ∈ CO ) is not differentiable, a similar
approach that uses subgradients (instead of gradients) enables
one to derive the following closed-form expression [30]:



X
1
ẑu = projCO 
hH y −
hj zj .
(14)
khu k22 u
j6=u
Here, projCO (·) is the orthogonal projection onto the convex
polytope CO and is given by
w
if w ∈ CO
projCO (w) =
(15)
arg minq∈CO |w − q| if w ∈
/ CO .
In words, if the argument w ∈ C is within the set CO , then the
projection outputs w; if w is outside the set CO , the projection
outputs the value q that is closest to w within the set CO in
terms of the Euclidean distance. We emphasize that for many
practically-relevant constellation sets O, the projection (15)
can be carried out efficiently. For any QAM constellation,
for example, we independently clip the real and imaginary
part of w onto the interval [−α, +α], where α is the radius
of the tightest box that covers the QAM constellation (see
Section II-D for the details). For BPSK with O = {−1, +1},
we clip the real part of w onto the interval [−1, +1] and set
the imaginary part to zero.5
5 Orthogonal projections for PSK constellations sets are also possible. The
development of efficient algorithms for PSK systems is left for future work.
Algorithm 1 Optimized Coordinate Descent (OCD)
1: inputs: y, H, and N0
2: initialization:
3:
r = y and z(0) = 0U ×1
4:
MMSE mode: α = N0 and C = C
5:
BOX mode: α = 0 and C = CO
6: preprocessing:
2
−1
7:
d−1
, u = 1, . . . , U
u = (khu k2 + α)
−1
2
8:
pu = du khu k2 , u = 1, . . . , U
9: equalization:
10: for k = 1, . . . , K do
11:
for u = 1, . . . , U
do
(k)
(k−1)
H
12:
zu = projC d−1
u hu r + pu zu
(k)
(k)
(k−1)
∆zu = zu − zu
(k)
r ← r − hu ∆zu
end for
16: end for
(K)
(K)
17: outputs: s̃ = [z1 , . . . , zU ]T
13:
14:
15:
B. Optimized Coordinate Descent (OCD)
Instead of blindly computing the updates (13) and (14)
for MMSE and BOX equalization, respectively, we perform
preprocessing and algorithm restructuring in order to minimize
the amount of (recurrent) operations during each of the k =
1, . . . , K iterations. These optimizations entail no performance
loss, i.e., both methods, OCD and CD, deliver exactly the same
results. We refer to the resulting method as the optimized CD
algorithm (short OCD), which is summarized in Algorithm 1.
OCD supports both BOX and MMSE equalization and the
individual optimization steps are as follows.6
1) Preprocessing: To reduce the computational complexity,
OCD precomputes certain key quantities that can be re-used
during each of the k = 1, . . . , K iterations. This preprocessing
step not only results in significant complexity savings during
the iterative process (compared to CD), but also simplifies
our hardware implementation (see Section IV). In particular,
we precompute so-called (regularized) inverse squared column
2
−1
norms of H, i.e., d−1
for u = 1, . . . , U ,
u = (khu k2 + α)
2
with α ≥ 0, as well as regularized gains pu = d−1
u khu k2 for
u = 1, . . . , U . In MMSE mode, the regularization parameter is
given by α = N0 ; in BOX mode, the regularization parameter
is given by α = 0, which yields pu = 1, u = 1, . . . , U .
2) Equalization: In order to avoid recurrent operations
during the equalization process, OCD performs incremental
updates and re-uses intermediate quantities during each of the
k = 1, . . . , K iterations. In essence, we perform sequential
updates on the so-called residual approximation vector, which
is defined as
r=y−
U
X
(k)
hj zj
(16)
j=1
6 The OCD algorithm proposed in the conference version of this paper [1]
differs from the one presented here. The operations in OCD as proposed
here have been restructured in order to (i) support MMSE as well as BOX
equalization and (ii) reduce the hardware complexity.
5
100
10−1
MMSE detector
OCD, K = 4
OCD fp, K = 4
GS, K = 4
CG, K = 4
10−2
2
3
4
5
6
Eb /N0 [dB]
7
100
MMSE detector
OCD, K = 3
OCD fp, K = 3
GS, K = 3
CG, K = 3
OCD, K = 2
GS, K = 2
CG, K = 2
Neumann, K = 3
10−1
10−2
−2
8
(a) 32 BS antennas and 8 users.
Packet error rate (PER)
Packet error rate (PER)
Packet error rate (PER)
100
−1
0
1
2
MMSE detector
OCD, K = 3
OCD fp, K = 3
GS, K = 3
CG, K = 3
OCD, K = 2
GS, K = 2
CG, K = 2
Neumann, K = 3
10−1
10−2
−4
3
Eb /N0 [dB]
(b) 64 BS antennas and 8 users.
−3
−2
−1
0
1
2
Eb /N0 [dB]
(c) 128 BS antennas and 8 users.
Fig. 1. Packet error rate (PER) for a massive MU-MIMO-OFDM system (“fp” denotes fixed-point performance). Optimized coordinate descent (OCD) with
box-constrained equalization achieves close-to-MMSE PER performance and outperforms the other three approximate equalization methods [7], [10], [13].
100
10−1
100
10−1
MMSE detector
OCD BOX, K = 3
OCD MMSE, K = 3
OCD BOX, K = 2
OCD MMSE, K = 2
Exact Inversion
OCD BOX, K = 4
OCD MMSE, K = 4
10−2
2
3
4
5
6
Eb /N0 [dB]
(a) 32 BS antennas and 8 users.
7
8
Packet error rate (PER)
Packet error rate (PER)
Packet error rate (PER)
100
10−2
−2
−1
0
1
2
Eb /N0 [dB]
(b) 64 BS antennas and 8 users.
3
MMSE detector
OCD BOX, K = 3
OCD MMSE, K = 3
OCD BOX, K = 2
OCD MMSE, K = 2
10−1
10−2
−4
−3
−2
−1
0
1
2
Eb /N0 [dB]
(c) 128 BS antennas and 8 users.
Fig. 2. Packet error rate (PER) for a massive MU-MIMO-OFDM system. BOX equalization outperforms MMSE equalization, especially for systems with a
smaller BS-to-user antenna ratio. Furthermore, both approximate equalization methods achieve near-exact MMSE performance for a small number of iterations.
at every algorithm iteration k = 1, . . . , K and for each user
u = 1, . . . , U . Note, however, that we do not recompute this
residual approximate vector for every iteration and user from
scratch. In contrary, we update the residual approximation
vector in every iteration and for each user by first computing
(k)
the symbol estimates zu on line 12 of Algorithm 1. We then
(k)
compute a so-called delta value ∆zu on line 13, which enables
us to update the residual r on line 14 without calculating the
residual (16) explicitly.
As mentioned above, OCD delivers exactly the same results
as CD, but does so at significantly lower computational
complexity. In fact, the original CD algorithm in Section III-A
requires one complex-valued inner product and U − 1 complex
scalar-by-vector multiplications per iteration k, whereas the
proposed OCD algorithm requires only one inner product and
one complex scalar-by-vector multiplication. More precisely,
for MMSE equalization, CD requires 4BU 2 + 2U real-valued
multiplications7 per iteration k, whereas OCD requires only
8BU + 4U real-valued multiplications. Hence, for a large
7 We
count 4 real-valued multiplications per complex-valued multiplication.
number of BS antennas B, OCD requires roughly U/2 times
lower complexity than CD per iteration.
C. LLR Approximation for OCD
To compute the LLR values (9) for MMSE and BOX
equalization using OCD, we must resort to an approximation
as we never explicitly compute the inverse A−1
w . To this
end, we use the approximation put forward in [11], [22]
for SC-FDMA-based systems. For OFDM, this approach
simplifies significantly and corresponds to approximating the
−1
channel gains by µ̃w,i = d−1
w,i gw,i , where dw,i is the ith
regularized inverse squared column norm of Hw and gi,w
is the entry in the ith main diagonal of the Gram matrix Gw
at subcarrier w. Furthermore, the approach from [11], [22]
applied to OFDM systems results in the following SINR
approximation: ρ̃w,i = µ̃w,i /(1 − µ̃w,i ). We refer the interested
reader to [22] for more details. As we will show next, this LLR
approximation enables near-optimal performance in massive
MU-MIMO systems with large BS-to-user-antenna ratios.
6
D. Error-Rate Performance
In order to assess the error-rate performance for the proposed
OCD-BOX algorithm, we perform Monte-Carlo simulations
in a coded 20 MHz MIMO-OFDM uplink system with 2048
subcarriers, where 1200 are used for data transmission as in
LTE Advanced (LTE-A) [31]. We use 64-QAM with Gray
mapping and a rate-3/4 turbo code. To account for spatial and
frequency correlation, we generate channel matrices using the
WINNER-Phase-2 model [32] with 7.8 cm antenna spacing as
in [11], [22]. For channel decoding, we use a log-MAP turbo
decoder. We report the packet error-rate, which is obtained by
coding over one OFDM symbol with 1200 data subcarriers.
The signal-to-noise-ratio (SNR) per bit in decibels, defined as
!
E ksk2
Eb
10 log10
= 10 log10
.
(17)
N0
QE[knk2 ]
can be configured in terms of the numbers of supported users U
and maximum number of iterations K.
A. Architecture Overview
Figure 3 shows two high-level block diagrams of the proposed OCD architecture. The inputs of our architecture are
the channel matrix Hw , the residual error vector r (which is
initialized to the received vector yw ), and the regularization
parameter α, which we initialized to N0 and 0 for MMSE and
BOX equalization, respectively. Our architecture supports two
operation modes: (a) preprocessing (lines 6–8 of Algorithm 1)
and (b) OCD-based qualization (lines 10–16). Preprocessing
and equalization are carried out in a B-wide vector pipeline,
i.e., we process B-dimensional vectors at a time. In the
preprocessing mode, we compute the regularized inverse
squared column norms d−1
u , u = 1, . . . , U , as well as the
regularized gains pu , u = 1, . . . , U . In the equalization mode,
Figures 1 and 2 compare the packet error rate (PER) for we perform the iterations on lines 12–13 of Algorithm 1. In
OCD-BOX with other exact and approximate data-detection order to support these two operation modes without the need of
methods for massive MU-MIMO systems with various antenna redundant computation units, the processing pipeline shares the
configurations. In particular, we show PER results for Neumann- key building blocks used in both modes. In particular, both of
series detection [7], CG-based detection [10], and Gauss-Seidel the supported modes share the inner-product unit and the right(GS)-based detection [13]. We also include an exact linear shift unit (highlighted in red in Figure 3). The inner product
MMSE equalizer as a reference. For all considered antenna unit consists of B parallel complex-valued multipliers followed
configurations, OCD-BOX outperforms Neumann, CG, and GS by a balanced adder tree. We use multiplexers at the input of
detection for the same iteration count. We see that OCD with the inner product unit, which enables us to switch between
BOX equalization (OCD-BOX for short) achieves near-exact preprocessing and equalization on a per-clock cycle basis.
MMSE performance for only three iterations (K = 3) for
One of the main implementation challenges of the proposed
64 and 128 BS antennas, whereas K = 4 is required for the OCD algorithm are data dependencies between successive
“not-so-large” system with 32 BS antennas; lower values of K iterations, which prevent traditional architecture pipelining.
result in a high error floor. These results confirm that for larger In particular, as it can be seen on line 14 of Algorithm 1,
BS-to-user-antenna ratios, approximate linear data detectors each OCD iteration updates the temporary vector r and the
approach the performance of the MMSE detector. We note vector z(k+1) given the previous vectors r and z(k) . Hence,
u
u
that for the considered antenna configurations, linear MMSE in order to achieve high throughput, we deploy pipeline
detection achieves near-ML performance [7].
interleaving [33], i.e., we simultaneously process multiple
Figures 2(a), 2(b), and 2(c) compare the PER for OCD-BOX subcarriers in a parallel and interleaved manner within the
against OCD with MMSE equalization (short OCD-MMSE). same architecture. For example, after performing an OCD
The performance of OCD-BOX is superior than that of OCD- iteration for the first subcarrier, we start an OCD iteration for
MMSE, especially in the 32 BS antenna, 8 user case. In general, the second subcarrier in the next clock cycle; we repeat this
the performance difference is more pronounced for smaller interleaving process until all pipeline stages are fully occupied.
BS-to-user-antenna ratios. This observation is in accordance to Our final architecture uses a total number of 24 pipeline stages,
recent theoretical results [23], and can be addressed to the fact which enables our design to achieve up to 260 MHz in a Xilinx
that the box constraint around the constellation is more accurate Virtex-7 FPGA (see Section V for more details). We note
than the quadratic penalty g MMSE (z) = N0 kzk22 imposed by that it is possible to achieve even higher clock frequencies
MMSE equalization.
by increasing the number of pipeline stages (especially for
We conclude by noting that for many modern wireless smaller small B); this approach, however, results in a significant
communication standards (such as LTE-A [31]) achieving a hardware overhead.
target PER of 10% is sufficient. The proposed OCD detector
is able to meet this target performance at only a small SNR B. Architecture and Fixed-point Optimization
loss compared to the exact MMSE-based data detector.
In order to optimize the hardware efficiency of our architecture, we use fixed-point arithmetic throughout our design.
IV. VLSI A RCHITECTURE
We achieved a negligible implementation loss with 16 bit
We now detail our VLSI architecture for OCD-based precision with 11 fractional bit for most internal signals;
MMSE and BOX equalization. The architecture was designed see Figure 1 for the fixed-point (fp) performance. Our design
and optimized using Xilinx Vivado HLS (version 2015.2), has an implementation loss of less than 0.2 dB SNR (measured
which allows us to conveniently simulate, parameterize, and at a target PER of 10%) compared to floating-point performance
generate different OCD designs that support various antenna for the considered scenarios, which is a result of the following
configurations at design time. At run-time, the proposed designs two optimizations.
7
௨
(Bx32)b
௨
ு
௨ ௨
16b
72b
72b
>>
log2(B)
16b
·
௨ ିଵ 16b
/
+
x
௨
16b
(a) OCD preprocessing mode.
!
16b
$
(Bx32)b
"#$%&
72b
#'
!$
>>
log2(B)
"#&
#'
!
32b
x
+
proj
x
16b
"#$%&
update "!
"
"#$%&
32b
"!
"#&
"#&
"!
32b
"!
32b
32b
"!
(Bx32)b
32b
+
∆"!
32b
#! ∆"!#
32b
$
-
(Bx32)b
+
$
(Bx32)b
(Bx32)b
update !
(b) OCD iteration mode.
Fig. 3. High-level block diagram of the proposed OCD-based preprocessing and equalization pipeline. The pipeline is reconfigurable for various BS-antenna
configurations at design time, and is able to perform preprocessing as well as MMSE or BOX equalization. The shared computation units between preprocessing
and equalization are highlighted in red.
1) Inner-product unit: This unit first computes entry-wise
products of two B-dimensional vectors and then, generates the
final sum of these products. We use a balanced adder tree to
compute the final sum and 36 bit adders to achieve sufficiently
high arithmetic internal precision. During preprocessing, the
inner-product unit computes khu k22 (line 7 of Algorithm 1);
during equalization, the same unit computes hH
u (r) (line 12).
As both of these terms are close to B (for large values of
B), we shift these terms by b = dlog2 (B)e bits to the right
in order to reduce the dynamic range. Since we shift khu k22
by b to the right, when we compute the reciprocal value,
d−1
= (khu k22 + α)−1 , we effectively shift the reciprocal
u
value d−1
u by b bits to the left. In the inner-product unit, we
also shift the term hH
u r by b bits to the right. Consequently,
we do not need to undo both of these shifts, as they cancel
out during the multiplication on line 12 of Algorithm 1.
2) Reciprocal unit: This unit consists of two parts. The first
part normalizes the input value to the range [0.5, 1], which is
accomplished using a leading-zero detector and programmable
shift to the left. The second part generates a reciprocal value
for the normalized input using a look-up table (LUT). We use
a FPGA BRAM18 to implement a 18 bit, 2048 entry LUT,
where the leading 11 bits of the normalized input value are
TABLE I
I MPLEMENTATION RESULTS ON A X ILINX V IRTEX -7
XC7VX690T FPGA FOR DIFFERENT BS ANTENNA NUMBERS
Array size
B = 32
B = 64
B = 128
#
#
#
#
#
2 873
6 059
10 704
198
2
6 508
12 588
24 801
390
2
11 094
23 914
43 008
774
2
261 MHz
261 MHz
258 MHz
of
of
of
of
of
Slices
LUTs
FFs
DSP48s
BRAM18s
Max. clock frequency
used to point to the entry in the LUT that stores the associated
normalized reciprocal. Finally, we denormalize the normalized
reciprocal value by another left shift.
V. I MPLEMENTATION R ESULTS AND C OMPARISON
We now show FPGA implementation results and compare
our design to the recently proposed data-detectors for massive
MU-MIMO systems in [7], [10], [12], [13].
8
Main units
# of Slices
# of LUTs
# of FFs
# of DSP48s
# of BRAM18s
B = 32
r update unit
Inner-product unit
hu ∆zu scaling unit
Miscellaneous
Total
256 (8.91%)
811 (28.2%)
265 (9.22%)
1 541 (53.6%)
2 873 (100%)
1 024 (16.9%)
1 045 (17.3%)
249 (4.11%)
3 741 (61.74%)
6 059 (100%)
0 (0%)
2 416 (22.6%)
1 137 (10.6%)
7 151 (66.8%)
10 704 (100%)
0 (0%)
96 (48.5%)
96 (48.5%)
6 (3.0%)
198 (100%)
0
0
0
2
2
(0%)
(0%)
(0%)
(100%)
(100%)
B = 64
r update unit
Inner-product unit
hu ∆zu scaling unit
Miscellaneous
Total
512 (7.87%)
1 627 (25.0%)
485 (7.45%)
3 884 (59.7%)
6 508 (100%)
2 048 (16.3%)
2 006 (15.9%)
505 (4.01%)
8 029 (63.8%)
12 588 (100%)
0 (0%)
5 776 (23.3%)
2 161 (8.71%)
16 864 (68.0%)
24 801 (100%)
0 (0%)
192 (49.2%)
192 (49.2%)
6 (1.6%)
390 (100%)
0
0
0
2
2
(0%)
(0%)
(0%)
(100%)
(100%)
B = 128
TABLE II
A REA BREAKDOWN ON A X ILINX V IRTEX -7 XC7VX690T FPGA FOR DIFFERENT BS ANTENNA NUMBERS
r update unit
Inner-product unit
hu ∆zu scaling unit
Miscellaneous
Total
1 024 (9.23%)
3 447 (31.1%)
1 955 (17.6%)
4 668 (42.1%)
11 094 (100%)
4 096 (17.1%)
4 109 (17.2%)
5 120 (21.4%)
10 589 (44.3%)
23 914 (100%)
0 (0%)
11 676 (27.0%)
4 211 (9.72%)
27 421 (63.3%)
43 308 (100%)
0 (0%)
384 (49.6%)
384 (49.6%)
6 (0.8%)
774 (100%)
0
0
0
2
2
(0%)
(0%)
(0%)
(100%)
(100%)
with respect to U . For example, doubling U doubles the number
of bits per subcarrier. However, since the number of OCD
updates is KU , the number of required clock cycles also
doubles; this results in a constant throughput. For K = 3
K=1
K=2
K=3
K=4
iterations, which was shown in Figure 1 to achieve near-optimal
Max. throughput [Mb/s] 1 363
496
376
302
performance, our design achieves 376 Mb/s. Hence, the use of
Latency [µs]
1.58
2.33
3.08
3.82
only three parallel instances (to process subcarriers in parallel)
would easily exceed 1.1 Gb/s, while consuming less than 65%
of the FPGA’s BRAM18s (cf. Table IV).
A. FPGA Implementation Results
The processing latency increases roughly linearly with
We designed three different implementations for the fol- respect to K and U . More specifically, the processing latency
lowing BS antenna configurations: B = 32, B = 64 and of this design is approximately 24(K + 1)U + O clock cycles,
B = 128. For each configuration, we provide post place-and- where O is the number of cycles required to flush the pipeline.
route implementation results on a Xilinx Virtex-7 XC7VX690T Typically, 26 cycles are required to flush the pipeline, the exact
FPGA. All our designs support U ≤ 32 users and K ≤ 256 value of O depends on B. The (approximately) linear increase
OCD iterations; both of these parameters can be set at run-time. in K can be seen in Table III and for K = 3, our design
The hardware complexity, resource utilization, and maximum requires only 3.08 µs to produce its first equalized output.
clock frequency results are summarized in Table I. We note
that there is no particular critical path in all our designs as B. Comparison
Vivado HLS evenly optimizes the delays among all pipeline
Table IV compares OCD to other, recently proposed largestages. A detailed area breakdown of the main units is shown in
scale
MIMO data detectors, namely the conjugate gradient
Table II. The “r update unit” corresponds to the output adder
(CG)-based
detector [10], the Neumann-series detector [7], the
in Figure 3(b); the “inner-product unit” corresponds to the unit
Gauss-Seidel
(GS) detector [13], and triangular approximate
H
H
that computes hu hu and hu r in Figure 3(a) and Figure 3(b),
semidefinite
relaxation
(TASER) [12]. All of these detectors
respectively; the “hu ∆zu scaling unit” corresponds to the
have
been
implemented
on the same FPGA and for a 128
scaling block in Figure 3(a); all remaining circuitry has been
BS
antenna,
8
user
system.
We see that for the same system
flattened by Vivado HLS and is subsumed in “miscellaneous.”
configuration,
OCD
outperforms
all other designs in terms of
Since the proposed architecture performs operations on Bhardware
efficiency,
which
we
define
as throughput per FPGA
dimensional vectors, the resource utilization (excluding the
LUTs.
Furthermore,
our
OCD
detector
achieves superior PER
BRAMs) scales linearly with B. Since the quantities Hw
performance
than
the
CG,
Neumann,
and GS detector (see
and yw are assumed to be stored in external memories, our
Figs.
1(b)
and
1(c)),
which
demonstrates
the effectiveness
OCD architecture only uses two BRAM18s: one for the
of
OCD.
TASER,
in
contrast,
achieves
better
error-rate perreciprocal LUT and one to store the regularized channel
8
formance
for
the
considered
antenna
configuration
but only
gains pu , u = 1, . . . , U .
supports QPSK constellations. We note that the throughput of
The maximum achievable throughput as well as the process(approximate) linear detectors, such as the ones in [7], [10], [13]
ing latency are shown in Table III. We see that the throughput
scales linearly in the number of bits Q per symbol; for TASER,
only depends on the maximum iteration number K and the
however, the throughput is limited by QPSK modulation, which
clock frequency, but does not depend on U . The reason is
because the number of bits per subcarrier and the number of
8 TASER achieves near-ML performance in “not-so-massive” MIMO systems,
clock cycles required to process 24 subcarriers grows linearly where the number of users is comparable to the number of BS antennas.
TABLE III
T HROUGHPUT AND LATENCY ON A X ILINX V IRTEX -7 XC7VX690T FPGA
FOR K ITERATIONS AND 64-QAM, AND 128 BS AND 8 USER ANTENNAS
9
TABLE IV
C OMPARISON OF 128 × 8 DATA DETECTORS FOR MASSIVE MU-MIMO SYSTEM ON A X ILINX V IRTEX -7 XC7VX690T FPGA
a The
Detector
CG [10]
Neumann [7]
Gauss-Seidel [13]
TASER [12]
OCD
Performance
Highest modulation
Iteration count K
near-MMSE
64-QAM
3
near-MMSE
64-QAM
3
near-MMSE
64-QAM
1a
near-ML
QPSK
3
near-MMSE
64-QAM
3
#
#
#
#
#
1 094 (1.0%)
3 324 (0.8%)
3 878 (0.4%)
33 (0.9%)
1
48 244 (45%)
148 797 (34%)
161 934 (19%)
1 016 (28%)
16
n.a.
18 976 (4.3%)
15 864 (1.8%)
232 (6.3%)
6
4 350 (4.0%)
13 779 (3.2%)
6 857 (0.8%)
168 (5.7%)
0
11 094 (10%)
23 914 (5.5%)
43 008 (4.96%)
774 (21.5%)
2
Maximum clock frequency [MHz]
Latency [clock cycles]
Maximum throughput [Mb/s]
412
951
20
317
196
621
309
n.a.
48
225
72
50
258
795
376
Throughput/LUTs
6 017
4 173
2 530
3 629
15 597
of
of
of
of
of
slices
LUTs
FFs
DSP48s
BRAM18s
method uses a special Neumann-series initializer followed by one GS iteration.
prevents this detector to achieve comparable throughputs as
the other approximate methods.
In summary, we see that OCD outperforms the next-best
design (namely the CG-detector from [10]) by more than 2.6×
in terms of hardware efficiency. The reasons for this advantage
are due to the facts that (i) OCD can be implemented in a very
regular and parallel manner and (ii) preprocessing requires
significantly lower complexity compared to that of the other
detectors that require the computation of the regularized Gram
matrix Aw , which can be a significant burden in massive
MU-MIMO-OFDM systems.
VI. C ONCLUSIONS
We have proposed a novel coordinate descent (CD)-based
data detector, called optimized CD (OCD), for massive MUMIMO systems that use orthogonal frequency division multiplexing (OFDM). The proposed OCD detector enables highperformance linear MMSE and non-linear box-constrained
data detection using a simple, parallel VLSI architecture that
requires low hardware complexity. Our FPGA reference design
achieves 376 Mb/s for a 128 BS antenna, 8 user system, and
substantially outperforms existing approximate linear datadetection methods in terms of hardware efficiency and/or errorrate performance. Our results show that OCD enables realistic
OFDM-based massive MU-MIMO systems to support tens of
users communicating with hundreds of BS antennas, while
achieving high throughput at low implementation costs.
There are many avenues for future work. OCD can also
be used for linear and non-linear precoding in the massive
MU-MIMO downlink; a corresponding study is part of ongoing
work. Computing exact soft-output values for OCD-based
detection (for MMSE and BOX equalization) is an interesting
open research problem. Finally, accelerated CD algorithms have
been proposed recently [34]; such methods may lead to even
faster convergence and hence, could enable higher throughput
at the same error-rate performance when implemented in VLSI.
VII. ACKNOWLEDGMENTS
C. Studer would like to thank Tom Goldstein, Charles Jeon,
Shahriar Shahabuddin for insightful discussions on the box-
constrained equalization method. The work of M. Wu and J.
R. Cavallaro was supported in part by Xilinx Inc., and by the
US National Science Foundation (NSF) under grants ECCS1408370, CNS-1265332, and ECCS-1232274. The work of
C. Studer was supported in part by Xilinx Inc. and by the US
NSF under grants ECCS-1408006 and CCF-1535897.
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